A Plactic Algebra Action on Bosonic Particle Configurations

We study an action of the plactic algebra on bosonic particle configurations. These particle configurations together with the action of the plactic generators can be identified with crystals of the quantum analogue of the symmetric tensor representations of the special linear Lie algebra $\mathfrak {s} \mathfrak {l}_{N}$ s l N . It turns out that this action factors through a quotient algebra that we call partic algebra, whose induced action on bosonic particle configurations is faithful. We describe a basis of the partic algebra explicitly in terms of a normal form for monomials, and we compute the center of the partic algebra.

Gröbner-Shirshov Bases for Plactic Algebras

A finite Gröbner-Shirshov basis is constructed for the plactic algebra of rank 3 over a field K. It is also shown that plactic algebras of rank exceeding 3 do not have finite Gröbner-Shirshov bases associated to the natural degree-lexicographic ordering on the corresponding free algebra. The latter is in contrast with the case of a strongly related class of algebras, called Chinese algebras.

THE RULE OF LITTLEWOOD-RICHARDSON IN A CONSTRUCTION OF BERENSTEIN-ZELEVINSKY

The rule of Littlewood-Richardson gives the decomposition of a product of Schur functions in the basis of the same functions. Each coefficient of this decomposition is the number of factorizations of a tableau of Yamanouchi in the plactic algebra. A. D. Berenstein and A. V. Zelevinksy prove that these coefficients are also the numbers of certain configurations called triangles. This text gives an explicit bijection between these triangles and the words of Yamanouchi.